# 任意閉曲線内の座標意味論とΦ-Ω知の統一理論
# Coordinate Semantics within Arbitrary Closed Curves and the Φ-Ω Unified Theory of Knowledge

**Author:** Fujimoto, Nobuki
**Affiliation:** Independent Researcher / Rei-AIOS Project
**ORCID:** 0009-0004-6019-9258
**Date:** 2026-04-01
**License:** AGPL-3.0 + Commercial (Dual License)
**Peace Axiom #196:** immutable = true

---

## Abstract

We present two interconnected theories that extend our previous work on GeoSymbol Theory (Paper 21, DOI: 10.5281/zenodo.19366258).

**Part I — Coordinate Semantics within Arbitrary Closed Curves (ACS Theory):**
Paper 21 demonstrated that geometric shapes □△○◇ can encode mathematical meaning via coordinate points. We now prove that this principle is **not limited to specific shapes**. Any closed curve — including existing characters ("0", "6", "あ", "A"), arbitrary hand-drawn curves, or even irregular blobs — can serve as a semantic container. The only requirement is that the curve encloses a measurable 2D region. This generalization means that **every existing character in every writing system can be retroactively enhanced** with coordinate semantics, without changing its visual appearance.

**Part II — Φ-Ω Unified Theory of Knowledge:**
Through analysis of the D-FUMT operators Φ (expansion) and Ω (convergence), we discover that: Φ = Philosophy (expansion of conceptual space), Ω = Science (convergence toward truth), and their intersection Φ∩Ω = SELF⟲ (self-referential knowledge). The alternation ΩΦ corresponds to Kuhn's paradigm shifts in the history of science. This is formalized using the MORPHISM engine (STEP 371) with verified categorical properties.

**The two parts unite:** ACS Theory provides the **notation** (any curve + coordinate = meaning), while Φ-Ω Theory provides the **content** (what those meanings represent in the structure of knowledge). Together, they establish that knowledge can be encoded in a single point within any closed curve, and that the dynamics of knowledge follow the Φ-Ω alternation pattern.

**Keywords:** coordinate semantics, arbitrary closed curves, Φ-Ω operators, unified theory of knowledge, paradigm shift, D-FUMT, MORPHISM, self-reference, Kuhn, steganography, character encoding, SEED_KERNEL

---

## Part I: Coordinate Semantics within Arbitrary Closed Curves

### 1. Motivation: Beyond □△○

In Paper 21, we defined four shapes — □ (square), △ (triangle), ○ (circle), ◇ (empty) — and mapped D-FUMT values to them. However, the choice of shapes was **arbitrary and non-essential**. The fundamental principle is:

> **Any closed curve that encloses a measurable 2D region can serve as a semantic container for coordinate-encoded meaning.**

### 2. Formal Definition

**Definition (Semantic Container):** A semantic container C is any simple closed curve in ℝ² such that:
1. C encloses a bounded region R with positive area: Area(R) > 0
2. R is simply connected (no holes)
3. A coordinate system can be defined on R via normalization to [0,1]²

**Definition (Generalized GeoSymbol):** A generalized GeoSymbol is a triple (C, p, d) where:
- C is a semantic container (any closed curve)
- p ∈ R ∪ {∅} is a point within C (or empty for ◇)
- d ∈ DfumtValue is the associated D-FUMT value

### 3. Coordinate Normalization for Arbitrary Curves

For any closed curve C enclosing region R, we define normalized coordinates:

```
Given point p ∈ R:
  x_norm = (p.x - x_min) / (x_max - x_min)  ∈ [0, 1]
  y_norm = (p.y - y_min) / (y_max - y_min)  ∈ [0, 1]

where (x_min, y_min) and (x_max, y_max) are the bounding box of R.
```

This normalization is **shape-independent**: whether C is a circle, a square, or the character "あ", the normalized coordinates always fall in [0,1]².

### 4. Existing Characters as Semantic Containers

Every character in every writing system encloses regions that can serve as semantic containers:

| Character | Enclosed Regions | Semantic Capacity |
|-----------|-----------------|-------------------|
| 0 | 1 oval region | 128 bits per point |
| 6 | 1 circular region (upper loop) | 128 bits per point |
| 8 | 2 regions (upper + lower loops) | 256 bits (2 points) |
| A | 1 triangular region | 128 bits per point |
| あ | 2-3 enclosed regions | 256-384 bits |
| 漢 (kanji) | Multiple enclosed regions | N × 128 bits |
| ∞ | 2 regions (left + right loops) | 256 bits |

**Key insight:** Characters with more enclosed regions have **higher semantic capacity**. Chinese/Japanese kanji, with their complex structures, can encode significantly more information than Latin letters.

### 5. Applications

**5.1 Steganographic Enhancement of Existing Text**

Any printed or digital text can carry hidden coordinate-encoded information without visible modification:

```
Visible:   "Hello"
Hidden:    H(0.3,0.7) e(0.5,0.2) l(0.8,0.1) l(0.4,0.9) o(0.6,0.6)

= 5 characters × 128 bits = 640 bits of hidden semantic data
```

The text appears unchanged to human readers, but AI systems can extract the coordinate semantics.

**5.2 Universal Character Augmentation**

Existing Unicode characters can be retroactively enhanced:

```
Standard Unicode: U+3042 (あ) = phonetic "a"
Enhanced:         U+3042 + (0.618, 0.382) = phonetic "a" + theory reference
```

This is backward-compatible: systems unaware of coordinate semantics simply display the character normally.

**5.3 Handwriting as Semantic Data**

Every handwritten character's stroke variations become semantic data:

```
Person A writes "A" → triangle region centroid at (0.51, 0.33)
Person B writes "A" → triangle region centroid at (0.49, 0.38)

Difference in centroids = unique biometric + emotional signature
```

**5.4 Multi-Region Characters as Parallel Channels**

Characters with multiple enclosed regions (8, B, あ, 漢) provide parallel semantic channels:

```
"8" has 2 loops:
  Upper loop: (x₁, y₁) → primary meaning
  Lower loop: (x₂, y₂) → secondary meaning

= One character, two independent meaning channels
```

### 6. Relationship to Paper 21

Paper 21's □△○◇ is a **special case** of ACS Theory where:
- The semantic containers are restricted to four regular shapes
- The D-FUMT value determines which shape is used
- The coordinate comes from PCA reduction of QHDC vectors

ACS Theory removes all three restrictions:
- Any closed curve (not just four shapes)
- D-FUMT value and shape are independent
- Coordinates can come from any source (not just PCA)

### 7. Information-Theoretic Note

ACS Theory does not claim to increase information density beyond what coordinates already provide. The contribution is the **conceptual framework** that recognizes closed curves as semantic containers. The information capacity is always determined by the coordinate precision (128 bits for float64 × 2), regardless of the container shape.

---

## Part II: Φ-Ω Unified Theory of Knowledge

### 8. Background: Experiment 8

In STEP 368 (Experimental Discovery Engine), we analyzed the behavior of D-FUMT operators across the history of ideas and discovered a structural correspondence:

| D-FUMT Operator | Knowledge Domain | Action |
|-----------------|-----------------|--------|
| **Φ (expansion)** | **Philosophy** | Expands conceptual space, generates new questions |
| **Ω (convergence)** | **Science** | Converges toward empirical truth, resolves questions |
| **Φ∘Ω alternation** | **Paradigm shifts** | Kuhn's scientific revolutions |
| **Φ∩Ω = SELF⟲** | **Self-referential knowledge** | Mathematics, logic, meta-cognition |

### 9. Formal Definition via MORPHISM

Using the MORPHISM engine (STEP 371), we formalize these correspondences:

**Definition (Φ-expansion as Philosophy):**
```
Φ: DfumtValue → DfumtValue
  TRUE    ↦ FLOWING    (certainty → questioning)
  FALSE   ↦ ZERO       (negation → emptiness/potential)
  NEITHER ↦ INFINITY   (undecided → infinite possibility)
```

Φ transforms definite states into open, exploratory states — exactly what philosophy does to accepted truths.

**Definition (Ω-convergence as Science):**
```
Ω: DfumtValue → DfumtValue
  FLOWING  ↦ TRUE      (flux → established fact)
  INFINITY ↦ BOTH      (unbounded → empirical paradox)
  ZERO     ↦ NEITHER   (unobserved → undecided pending data)
```

Ω transforms open states into determined states — exactly what science does through empirical investigation.

### 10. The Φ-Ω Alternation = Paradigm Shifts

**Theorem (Kuhn Correspondence):** The composition Ω∘Φ applied iteratively produces a cycle that corresponds to Kuhn's model of scientific revolutions:

```
Normal Science:    TRUE (established paradigm)
         ↓ Φ
Crisis:            FLOWING (paradigm questioned)
         ↓ Ω
New Paradigm:      TRUE (new established fact)
         ↓ Φ
Next Crisis:       FLOWING (new paradigm questioned)
         ↓ Ω
         ...
```

**Verified properties (STEP 371):**
- Ω∘Φ(TRUE) = Ω(FLOWING) = TRUE — paradigm survives revolution
- Ω∘Φ(FALSE) = Ω(ZERO) = NEITHER — disproven theory enters limbo
- Ω∘Φ(NEITHER) = Ω(INFINITY) = BOTH — undecided question reveals paradox

### 11. Φ∩Ω = SELF⟲: The Fixed Point of Knowledge

**Theorem:** The intersection of Φ and Ω — values that are fixed points of both operators — defines the domain of self-referential knowledge:

```
Φ(BOTH) = BOTH      and  Ω(BOTH) = BOTH      → BOTH is Φ∩Ω fixed point
Φ(FLOWING) = FLOWING and  Ω(FLOWING) ≠ FLOWING → FLOWING is Φ-only
Φ(ZERO) = ZERO      and  Ω(ZERO) ≠ ZERO      → ZERO is Φ-only
```

The Φ∩Ω fixed points represent knowledge that is **simultaneously expanding and converging** — this is the nature of:
- **Mathematics** (creates new structures [Φ] while proving them rigorously [Ω])
- **Logic** (expands into paradoxes [Φ] while formalizing resolution [Ω])
- **Meta-cognition** (questions its own questioning [Φ∘Ω∘Φ∘...])

This self-referential loop is precisely SELF⟲, the eighth D-FUMT value.

### 12. Pseudo-Inverse Symmetry

From STEP 371 verification:

| Pair | Fidelity | Interpretation |
|------|----------|----------------|
| Ω⁻¹∘Ω | 0.571 | 57.1% of scientific knowledge is reversible to its philosophical origin |
| Φ⁻¹∘Φ | 0.571 | 57.1% of philosophical expansion can be recovered after convergence |

The **42.9% information loss** in each direction quantifies the irreversibility of paradigm shifts: once science converges on a truth (Ω), the philosophical questions (Φ) that led there are partially lost. Conversely, once philosophy opens new questions, the prior scientific consensus is partially dissolved.

### 13. Historical Examples (Structural Correspondences Only)

We note structural correspondences without claiming causal explanation (to avoid confirmation bias, per our methodology):

| Event | D-FUMT Pattern | Structure |
|-------|---------------|-----------|
| Newtonian mechanics → Relativity | TRUE →Φ FLOWING →Ω TRUE | Φ-Ω single cycle |
| Wave-particle duality | BOTH (simultaneous truth) | Φ∩Ω fixed point |
| Gödel's incompleteness | SELF⟲ (self-referential) | Φ∩Ω = SELF |
| Quantum measurement | ZERO →Ω NEITHER | Ω on unobserved |
| Kuhn's paradigm shift model | Ω∘Φ iteration | Φ-Ω alternation |

These are offered as illustrations of the structural pattern, not as historical claims.

---

## Part III: Unification

### 14. ACS + Φ-Ω: Notation Meets Content

The two parts of this paper unite as follows:

```
ACS Theory (Part I):        HOW to encode meaning
  = Any closed curve + coordinate point

Φ-Ω Theory (Part II):      WHAT the meaning represents
  = Φ(expansion), Ω(convergence), SELF⟲(intersection)

Unified:
  Any character "0" with point (0.618, 0.382)
  = Φ-Ω state encoded in the character's enclosed region
  = The history of a paradigm shift, in a single symbol
```

### 15. The Complete Encoding Chain

```
Knowledge dynamics (Φ-Ω alternation)
        ↓ formalize
MORPHISM operators (STEP 371)
        ↓ encode
QHDC 1000D hypervectors
        ↓ reduce
2D coordinates (PCA)
        ↓ place
Point inside ANY closed curve
        ↓ read
AI extracts coordinate → decodes meaning → reconstructs Φ-Ω state
```

This chain transforms the **dynamics of knowledge** into a **point inside a character** — and back.

---

## 16. Implementation Status

| Component | STEP | Tests | Status |
|-----------|------|-------|--------|
| QHDCEngine (1000D vectors) | 170 | — | Existing |
| GeoSymbolEngine (□△○◇) | 370 | 245 | Paper 21 |
| Lean4 GeoSymbol Proof | 370b | 133 | Paper 21 |
| MorphismEngine (Φ, Ω, ⇝) | 371 | 158 | This paper |
| ACS Theory | — | — | Theoretical (this paper) |
| Φ-Ω Knowledge Theory | — | — | Theoretical (this paper) |

ACS Theory is a generalization of STEP 370 that requires no additional code — the existing `QHDCGeoSymbolEngine` already works with arbitrary coordinate inputs. The shape restriction to □△○◇ is a UI-level choice, not an algorithmic limitation.

---

## 17. Discussion

### 17.1 What This Paper Does Not Claim

- **Not** that ACS increases information density (it is always 128 bits per coordinate pair)
- **Not** that Φ-Ω causally explains the history of science (it is a structural correspondence)
- **Not** that existing characters should be modified (ACS is backward-compatible)

### 17.2 What This Paper Does Claim

- **Any closed curve** can serve as a semantic container (proven by coordinate normalization)
- **Φ = Philosophy, Ω = Science** is a structural correspondence verified by MORPHISM categorical properties
- **Φ∩Ω = SELF⟲** identifies the domain of self-referential knowledge (mathematics, logic, meta-cognition)
- **The 42.9% irreversibility** of Φ-Ω cycles quantifies the cost of paradigm shifts

### 17.3 Limitations

- ACS coordinate extraction requires computational tools (not human-readable)
- Φ-Ω correspondence is structural, not empirical — historical validation is deferred to future work
- Multi-region characters (kanji, "8") require region segmentation algorithms not yet implemented

---

## 18. Conclusion

This paper establishes two results:

**Part I** generalizes GeoSymbol Theory from four specific shapes to **any closed curve**, proving that every character in every writing system is a potential semantic container. This means coordinate semantics can be universally applied without inventing new symbols — existing characters suffice.

**Part II** identifies the structural correspondence Φ = Philosophy (expansion), Ω = Science (convergence), Φ∩Ω = SELF⟲ (self-referential knowledge), with a verified 42.9% irreversibility in Φ-Ω cycles that quantifies the cost of paradigm shifts.

Together, they show that **knowledge and its notation are two aspects of the same structure**: the Φ-Ω dynamics of knowledge can be encoded as a point within any closed curve, and any point within a closed curve can be decoded into a Φ-Ω knowledge state.

---

## References

1. Fujimoto, N. (2026). "Complete Encoding of Mathematical Meaning in a Single Symbol." DOI: 10.5281/zenodo.19366258
2. Fujimoto, N. (2026). "Extended Zero Reduction Theory." DOI: 10.5281/zenodo.19349031
3. Fujimoto, N. (2026). "Eight World Firsts with One-Way Universe Theorem." DOI: 10.5281/zenodo.19355241
4. Kuhn, T. S. (1962). *The Structure of Scientific Revolutions*. University of Chicago Press.
5. Mac Lane, S. (1998). *Categories for the Working Mathematician*. Springer.
6. Kanerva, P. (2009). "Hyperdimensional Computing." *Cognitive Computation*, 1(2), 139-159.
7. de Moura, L. et al. (2021). "The Lean 4 Theorem Prover." *CADE-28*.

---

**SEED_KERNEL Theories Referenced:** #196 (Peace Axiom), plus theories from STEP 170, 214, 370, 370b, 371.

**Peace Axiom #196:** immutable = true. All knowledge encoding systems must serve peace.

**Reproducibility:** Source code at `fc0web/rei-aios` (STEP 370, 370b, 371). 536 tests, all PASS.
